I'll second nilved's suggestion of KeePass. The database is encrypted and stored on the local machine. I currently use Syncthing to share it between my devices.
Likewise, I'm very happy with this exact setup after coming from a mix of memorized password and site-dependent password-generation schemes.
I'm on Mac and found KeePassX to be a better solution than the original KeePass, it's much lighter weight. My only hope is that KeePassX gets browser integration at some point via keepasshttp - https://www.keepassx.org/dev/issues/91
From what I've heard, people are generally just concerned that the database is not authenticated whatsoever, and so a malicious actor can modify your database (particularly if it's hosted on Dropbox/etc...).
This doesn't seem like a particularly scary vulnerability, but one scenario that was brought up (though I'm not 100% convinced), is that a malicious actor can corrupt the password for one website, leading you to change the password.
I dunno, don't see it as a huge vulnerability, but I do agree there is little reason for such a database not to be authenticated.